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Single-qubit gates based on targeted phase shifts in a 3D neutral atom array

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Science  24 Jun 2016:
Vol. 352, Issue 6293, pp. 1562-1565
DOI: 10.1126/science.aaf2581

How to single out the right atoms

For a quantum computer to be useful, its qubits have to be able to change their state in response to external stimuli. But when a large number of qubits are packed in a three-dimensional (3D) structure to optimize the use of space, altering one qubit can unintentionally change the state of others. Wang et al. devised a clever way to perform high-fidelity quantum gates only on intended qubits in a 3D array of Cs atoms. Although the operation initially changed the state of some of the other atoms, additional manipulation recovered their original state. The technique may be applicable to other quantum computing implementations.

Science, this issue p. 1562

Abstract

Although the quality of individual quantum bits (qubits) and quantum gates has been steadily improving, the number of qubits in a single system has increased quite slowly. Here, we demonstrate arbitrary single-qubit gates based on targeted phase shifts, an approach that can be applied to atom, ion, or other atom-like systems. These gates are highly insensitive to addressing beam imperfections and have little cross-talk, allowing for a dramatic scaling up of qubit number. We have performed gates in series on 48 individually targeted sites in a 40% full 5 by 5 by 5 three-dimensional array created by an optical lattice. Using randomized benchmarking, we demonstrate an average gate fidelity of 0.9962(16), with an average cross-talk fidelity of 0.9979(2) (numbers in parentheses indicate the one standard deviation uncertainty in the final digits).

The performance of isolated quantum gates has recently been improved for several types of qubits, including trapped ions (13), Josephson junctions (4), quantum dots (5), and neutral atoms (6). Single-qubit gate errors now approach or, in the case of ions, surpass the commonly accepted error threshold (error per gate < 10–4) (7, 8) for fault-tolerant quantum computation (912). It remains a challenge in all these systems to execute targeted gates on many qubits in close physical proximity to one another with fidelities comparable with those for isolated qubits (13, 14). Neutral atom and ion experiments have to date demonstrated the most qubits in the same system, 50 and 18 respectively (15, 17). The highest-fidelity gates in these systems are based on microwave transitions, but addressing schemes typically depend on either addressing light beams (6, 15, 1820), which are difficult to make as stable as microwaves, or magnetic field gradients (2, 21), which limit the number of addressed qubits. Here, we present a way to induce phase shifts on atoms at targeted sites in a 5 by 5 by 5 optical lattice that is highly insensitive to addressing laser beam fluctuations. We further show how to convert targeted phase shifts into arbitrary single-qubit gates.

In previous work, we performed single-site addressing in a three-dimensional (3D) lattice using crossed laser beams, to selectively induce ac Stark shifts in target atoms, and microwaves, to temporarily map quantum states from a field-insensitive storage basis to the Stark-shifted computational basis (15). Although we used most of the same physical elements in this work, the crucial difference is that the gates described here are based on phase shifts in the storage basis and do not require transitions out of it. Nonresonant microwaves are applied that give opposite-sign ac Zeeman shifts (analogous to ac Stark shifts, but for magnetic dipole transitions) for different atoms. A specific series of nonresonant pulses and global π-pulses on the qubit transition gives a zero net phase shift for nontarget atoms and a controllable net phase shift for target atoms. The resultant gate fidelity is much better than our previous gate because of this gate’s extreme insensitivity to the addressing beam alignment and power, the insensitivity of the storage basis to magnetic fields and vector light shifts, and the independence on the phase of the nonresonant microwave pulses.

Detailed descriptions of our apparatus can be found in (15, 22, 23). We optically trapped and reliably imaged neutral 133Cs atoms in a 5-μm spaced cubic optical lattice. The atoms were cooled to ~70% ground vibrational state occupancy and then microwave transferred into the qubit basis, the 6S1/2, Embedded Image, and Embedded Image hyperfine sublevels, which we will call Embedded Image and Embedded Image, respectively. Lattice light spontaneous emission is the largest source of decoherence, with a 7 s coherence time (Embedded Image) that is much longer than the typical microwave pulse time of 80 μs, which could be shortened with more microwave intensity. We detected the qubit states by clearing atoms in the Embedded Image states and imaging the Embedded Image atoms that remain.

To target an atom, we crossed at a right angle two circularly polarized, 880.250-nm addressing beams (beam waist, ~2.7 μm; Rayleigh range, ~26 μm). The addressing beams can be directed to a new target in <5 μs by using micro-electro-mechanical-system (MEMS) mirrors (24). The addressing beams induce only a modest ac Stark shift on the mF = 0 qubit states (~400 Hz), but they cause a vector light shift on the mF ≠ 0 levels. The vector light shifts are about twice as large for the target atom as for any other. This is illustrated in Fig. 1A, for which atoms are prepared in Embedded Image and a microwave near the Embedded Image transition is scanned. The resonances are visible for atoms at the intersection (orange, termed “cross” atoms), atoms in one addressing beam path (blue, termed “line” atoms), and the rest of the atoms (green, termed “spectator” atoms). The ac Stark shift for the line atoms, f, was chosen so that there is a region between the blue and orange peaks in which only a small fraction (2 × 10–4) of atoms in any class (cross, line, or spectator) makes the transition. When a microwave pulse is applied in that frequency range, atoms experience different ac Zeeman shifts depending on their class.

Fig. 1 Addressing spectrum, phase shift, and timing sequence.

(A) Spectra of the Embedded Image transition. We plot the ratio R of the number of detected F = 3 atoms to the initial number of atoms as a function of the phase-shifting microwave detuning from the non–ac-Stark–shifted resonance. There is a 570-mG bias field at 45° to the addressing beam axes. The green diamonds indicate spectators, the blue triangles indicate line atoms, and the orange circles indicate cross atoms. The solid lines are fits to Gaussians. The resonance frequencies of line atoms and cross atoms are marked by f and kf. (B) Exact calculation of the cumulative phase shift (φTarget) on a target atom as a function of the microwave detuning from the unshifted resonance. The optimum operation point is marked by δ0. Transitions out of the qubit basis are ignored. At δ0, these occur less than 2 × 10–4 of the times. (C) Addressing pulse sequence. (Top row) Blackman-profiled (to minimize off-resonant transitions) (35) microwave pulse sequence. The black pulses (80 μs) are resonant on the Embedded Image to Embedded Image transition and affect all atoms; the narrower pulses are π/2 rotations, which first create and then measure superpositions, and the wider pulses are π rotations, which give spin echoes. The purple pulses (120 μs) are addressing pulses detuned from the Embedded Image to Embedded Image transitions by δ0. (Second row from top) The addressing light intensity. The addressing light barely affects the trapping potential, so a linear ramp is optimal. It is effectively at full power for 252 μs. (Bottom two rows) Schematic of addressing beams in two planes. The atom color codes are as in (A).

The addressing pulse sequence for a pair of target atoms in two planes (Fig. 1C) consists of four stages (15). The qubit-resonant spin-echo pulses (Fig. 1C, black pulses on the microwave line) reverse the sign of the phase shifts, so that whatever phase shifts (ac Zeeman or ac Stark) a nontarget atom gets during the cross stages (Fig. 1C, stages 1 and 3) are exactly canceled by the shifts it gets during the dummy stages (Fig. 1C, stages 2 and 4), in which there is no cross atom. In contrast, the first target atom spends stage 1 as a cross, stage 3 as a spectator, and stages 2 and 4 as a line atom, and the second target atom spends stage 3 as a cross and stage 1 as a spectator. When the microwave frequency is chosen to be between the line and cross resonances, the change in the target atom’s status from cross to line changes the sign of the ac Zeeman shift. Away from the resonances, the net phase shift for the target atoms isEmbedded Image (1)where δ is the microwave detuning from the spectator resonance, k is the shift of cross atoms in units of f (16), Ω (typically 4 kHz) is the microwave Rabi frequency, and T is the pulse duration. Successive terms correspond to the integrated ac Zeeman shift on the first target atom during successive stages. The overall phase shift can be directly controlled by changing the power of the microwave field. The black curve in Fig. 1B shows the result of an exact calculation of the target atom’s phase shift as a function of δ. The phase shift minimum at δ0 = 74.9 kHz is the preferred operating point for the gate because in the vicinity of δ0, the shift depends quadratically on the change in δ and thus also on the addressing beams ac Stark shift, with the coefficient of 21 rad/(Δf/f)2. For example, a 2% change in f gives an 8-mrad phase shift, which in turn leads to only a 1 × 10–4 gate error. Because the intensity changes quadratically with beam alignment, the gate is sensitive to beam-pointing only at fourth order.

The phase shift on target sites amounts to a rotation about the z axis [an Rz(θ) gate], but a universal single-qubit gate requires arbitrary rotations about any arbitrary axis. We can make an Ry(θ) gate by combining the Rz(θ) gate with global Embedded Image rotationsEmbedded Image (2)For nontarget atoms, which see the global microwave pulses but experience no Rz(θ), Embedded Image clearly has no net effect. It is straightforward to generalize this formula to obtain arbitrary rotations on a Bloch sphere for target atoms. The corresponding complete set of single-qubit gates on target atoms all leave the nontarget atoms unchanged.

We have demonstrated one Rz(π/2) gate on a sequence of 48 randomly chosen sites (in 24 gate pairs) within a 5 by 5 by 5 array. Given the average initial site occupancy of ~40%, an average of 20 qubits experience the phase gate during each implementation, whereas 30 remain in their original quantum superposition. We probed the coherence of all the atoms by closing the spin echo sequence with a global π/2 pulse whose phase we scanned (Fig. 2). In Fig. 2, the open diamonds indicate nontarget atoms, and the solid circles indicate atoms at the 48 target sites. The corresponding curves are sinusoidal fits to the data. The dashed lines mark the maximum and minimum populations one expects given perfect gate fidelity, Embedded Image, defined as the square of the projection of the measured state onto the intended state, in the face of state preparation and measurement (SPAM) errors (16). From these curves, we determined that the error per gate pair, Embedded Image, is 25(13) × 10–4 for both target and nontarget atoms on average (numbers in parentheses indicate the 1 SD uncertainty in the final digits). At least for this particular gate, most of the error is clearly common to target and nontarget atoms. The good contrast of the target atom fringe illustrates the homogeneity of the phase shifts across the ensemble. The error per gate, Embedded Image, is thus 13(7) × 10–4 on average.

Fig. 2 Interference fringe of one Rz(π/2) phase gate.

The gate is applied to a succession of 48 randomly chosen sites (in pairs) in each implementation, and the data show averages over 10 implementations. The solid circles correspond to the target atoms, and the open diamonds correspond to nontarget atoms. Solid lines are fits to data. Dashed lines mark the maximum and minimum possible populations. After all these gates, the net fidelity Embedded Image for target atoms is 0.94(3), and for the nontarget atoms it is 0.94(2). The average error per gate for both target and nontarget atoms is 13(7) × 10–4.

To graphically illustrate the flexibility of our gates, we have performed an Rz(π) rotation on a 32-site pattern (Fig. 3). We rotated the superpositions of a sequence of pairs of target atoms from Embedded Image to Embedded Image and then used a π/2 detection pulse, phase-shifted by π, to return spectator atoms to Embedded Image and target atoms to Embedded Image. Images from five planes summed over 50 implementations are shown in Fig. 3.

Fig. 3 Rz(π) gates performed in a specific pattern.

“1” to “5” are fluorescent images of five successive planes. Each image is the sum of 50 implementations. For clarity, the contrast is enhanced by using the same set of dark/bright thresholds on all pictures to account for the shot noise. There are no targeted sites in planes 2 and 4; the light collected there is from atoms in the adjacent planes. (Bottom right) Illustration marking the targeted sites in blue and the untargeted sites in green, with plane shading and blue lines to guide the eye.

To confirm the robustness of this gate, we applied an Rz(π/2) gate to 24 targets and measured the probability that atoms return to Embedded Image after a π/2 pulse as a function of the fractional change in addressing beam intensity (Fig. 4, inset) (16). The data confirm the theoretical prediction (solid line) of extreme insensitivity to addressing beam intensity (Fig. 1B).

Fig. 4 Semi-log plot of fidelity from a randomized benchmarking sequence.

We plot the probability of returning the two target atoms to Embedded Image after a randomized benchmarking sequence versus the sequence length. At each length, we use three randomized CG sequences, each of which is combined with three sets of randomized PG sequences, for a total of nine points. Each data point is averaged over ~100 implementations. A least squares fit using the dif = 0.1128(68) determined from the auxiliary RB measurement on nontarget atoms gives Embedded Image (16). (Inset) Fidelity of an Rz(π/2) gate as a function of the fractional change of the addressing ac Stark shift. The points are experimental data, and the curve is from the exact calculation.

To fully characterize gate performance, we used the standard randomized benchmarking (RB) protocol (25), which has been used in nuclear magnetic resonance (26), quantum dots (5), ion systems (27), and neutral atoms systems (6, 28, 29). We implemented RB by choosing computation gates (CG) at random from Embedded Image and Pauli gates (PG) at random from {Rx(±π), Ry(±π), Rz(±π), I}, where I is the identity operation (30). After each randomized sequence, a detection gate is applied that in the absence of errors would return either the target qubits or the nontarget qubits to Embedded Image. For the targets, the average value of Embedded Image decays exponentially with the length of the randomized sequenceEmbedded Image (3)where dif/2 is the SPAM error and l is the number of CG-PG operations applied to the pair of target atoms. A similar expression yields Embedded Image and Embedded Image for the spectator and line atoms, respectively—the unwanted changes wrought on the nontarget atoms when a random series of gates is executed on the targets. In this way, we can characterize cross-talk and extract errors per gate from errors per gate pair. For target atoms, Embedded Image; for spectator atoms, Embedded Image; and for line atoms, Embedded Image.

First analyzing the nontarget data (fig. S2), we determined that dif = 0.11(1), Embedded Image , and Embedded Image. The cross-talk, defined as the average error per gate for nontarget atoms, is Embedded Image. Only global microwave pulse imperfections contribute to Embedded Image. Their contribution to Embedded Image may differ because the microwave sequence is optimized for spectators. Spontaneous emission from the addressing beams adds to Embedded Image, with a calculable contribution of 8 × 10–4. The RB data for the two target atoms are shown in Fig. 4. Using the dif from the nontargets, the fit of Eq. 3 to the data yields Embedded Image.

A summary of the errors for the target and nontarget atoms is shown in table S1. The errors can mostly be traced to microwave power stability (17 × 10–4 and 16 × 10–4). We expect that reaching the state of the art (2, 27), and perhaps tweaking our spin echo infrastructure (31) (fig. S1), can bring it below 10–4. The next largest contribution is from the readily calculable spontaneous emission of addressing light (16 × 10–4 for target atoms). Its contribution is invariant with gate times because f must be changed inversely with gate time. Addressing with light between the D1 (Embedded Image) and D2 (Embedded Image) lines, as we do, locally minimizes spontaneous emission, but doubling the wavelength would further reduce spontaneous emission by a factor of 8 without dramatically compromising site-addressing. The required powerful addressing beams would exert a substantial force on line atoms, but deeper lattices and adiabatic addressing-beam turn-on would make the associated heating negligible. In our current experiment, we need to wait 70 μs after the addressing beams are turned on for our intensity lock to settle. Technical improvement there could halve the time the light is on, ultimately leaving the spontaneous emission contribution to the error just below 10–4 per gate. That limit is based on addressing with a differential light shift. It would disappear were the same basic scheme to be used on a three-level system, in which only one of the qubit states is strongly ac Stark–shifted by the addressing light. In that case, the phase-shifting microwaves (or light) would simply be off-resonant from the qubit transition, and the error caused by spontaneous emission would decrease proportional to the detuning. Because the dominant target errors have the same sources as the nontarget errors, improvements to the gate will correspondingly improve the cross-talk. Adapting this method to more common 1D and 2D geometries is straightforward. Only a single addressing beam is needed, and the dummy stages would be executed with half-intensity addressing beams. The same insensitivity to addressing beam power and alignment would follow.

Scalable addressing is an important step toward a scalable quantum computer. Future milestones that must be passed on the road to scalable neutral atom quantum computation include scalable addressing for two-qubit gates (3234), reliable site filling (19), and the implementation of error correction (7).

Supplementary Materials

www.sciencemag.org/content/352/6293/1562/suppl/DC1

Materials and Methods

Figs. S1 to S3

Table S1

References (3639)

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
Acknowledgments: This work was supported by the U.S. National Science Foundation grant PHY-1520976. The data presented in this report are available on request to D.S.W.

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